Hey, one of my university professors! GF2^8 is also great for RAID6+ and Reed Solomon codes.

Back in the day it was tractable to build AES lookup tables for computing MixColumns and SubBytes similar to the site's multiplication table. Then djb found a remote timing attack, so processors started implementing hardware AES instructions.

> Inverses exist for each operation, which makes decryption possible.

It is worth mentioning that in modern AES usage this feature does not really matter, because block cipher is used in CTR mode. In CTR block cipher is used to create long keystream bytes (pretty much one-time-pad key). This keystream gets xor'ed with plaintext to get ciphertext. Then getting from ciphertext back to plaintext is exactly the same xor operation with same keystream, no inverse AES operations needed.

I don't know that "Weirdly" is right for the polynomial equivalence -- isn't that what Galois was trying to do in the first place (prove things about polynomials such as the impossibility of analytic solutions to higher order polynomials, and the impossibility of constructing a heptagon)?